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No they are not telling you the truth. You must sign a new lease every year.......if they fail to have you sign a new lease it reverts to month to month. Unless your original lease stipulated that the lease extends each year unless otherwise notified.
If this is the case you will want to notify them you will not be renewing the lease when it expires. Keep in mind they can request that you leave if you do not wish to sign a new lease. That is their option. If you leave prior to the lease ending you are held responsible for the remaining amount on the lease until it is rented and advertising to get it rented.